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A topological group is a group G equipped with a topology such that the group operations (multiplication and inversion) are continuous.

An open set in a topological space is a set that, around each of its points, contains a neighborhood entirely contained within the set.

A neighborhood of a point is a set that includes an open set containing the point.

A group is a set equipped with a single associative binary operation that has an identity element and where every element has an inverse.

A homeomorphism is a continuous function between topological spaces that has a continuous inverse function.

A topology on a set X is a collection of open sets that include X and the empty set, is closed under arbitrary union and finite intersection.

In group theory, the inverse of an element a is an element that, when combined with a under the group operation, yields the identity element.

A binary operation $*$ on a set is associative if $(a * b) * c = a * (b * c)$ for all elements $a$, $b$, and $c$ in the set.

In a group, the identity element is the element which, when combined with any element of the group, yields that element.

A Hausdorff space (or T2 space) is a topological space where for any two distinct points there exist disjoint open sets containing each of the points.

A map between topological spaces is continuous if the preimage of every open set is open.